Triangulated categories of logarithmic motives over a field
Triangulated categories of logarithmic motives over a field
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场上对数动机的三角类别
DOI：
10.24033/ast.1172
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
P. Ostvaer
中科院分区：
文献类型：
--
作者：
F. Binda;Doosung Park;P. Ostvaer
在这项工作中，我们开发了一个理论的动机，对数计划领域的意义上的方丹，Illusie和加藤。我们的建设是基于有限的日志对应的概念，划分Nisnevich拓扑的日志计划，和参数化同伦的想法$\overline{\square}$，即投影线相对于其紧化的对数结构在无穷远。霍奇上同调的对数计划是一个例子的$\overline{\square}$-不变理论是代表的范畴内的对数动机。这与Voevodsky的动机范畴和$\mathbb{A}^{1}$-不变理论相似。假设奇异性的解决方案，我们确定后者与完整的子范畴组成的$\mathbb{A}^{1}$-局部对象的对数动机的范畴。明显的属性，如$\overline{\square}$-同伦不变性，迈尔-Vietoris覆盖，和一个对称的monoidal结构见证了设置的鲁棒性。此外，我们证明了对数动机满足的基本性质，如一个射影丛定理，爆破区分三角形，和Gysin区分三角形。
In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log schemes, and the idea of parameterizing homotopies by $\overline{\square}$, i.e. the projective line with respect to its compactifying logarithmic structure at infinity. Hodge cohomology of log schemes is an example of an $\overline{\square}$-invariant theory that is representable in the category of logarithmic motives. This bears a resemblance to Voevodsky's category of motives and $\mathbb{A}^{1}$-invariant theories. Assuming resolution of singularities, we identify the latter with the full subcategory comprised of $\mathbb{A}^{1}$-local objects in the category of logarithmic motives. Palpable properties such as $\overline{\square}$-homotopy invariance, Mayer-Vietoris for coverings, and a symmetric monoidal structure witness the robustness of the setup. Moreover, we show that logarithmic motives satisfy fundamental properties such as a projective bundle theorem, a blow-up distinguished triangle, and a Gysin distinguished triangle.
群计划行动的动机同伦理论
影响因子：
1.1
作者：
J. Heller;A. Krishna;P. A. Østvær
通讯作者：
P. A. Østvær